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If someone was 137 and 5 foot 7 in. how many newtons would it take them to escape the gravity of earth?

First let me address that the force of gravity is a force that can never be escaped. The earth pulls on the even the faintest star that you can spot in the sky. However, an object can move fast enough so that it won’t fall back to earth until it is infinitely far away. In others words, an object moving beyond a critical speed will never come back. The critical speed is called the escape velocity. An object moving at escape velocity will continue to slow down as it travels through space but it will never come to rest. The escape velocity only depends on the mass and the radius of the planet from which it is escaping. Ironically, it doesn’t depend on the object that is moving. If you could run fast enough and leap, then you would be free from the earth. However, the escape velocity for the earth is 25,000 mph or 7 miles per second. There are only a few objects on earth that can travel that fast: Rockets!

Now, you asked how many Newtons it would take to launch someone so that they would achieve escape velocity. Newtons are a measure of force. This is similar to asking about the force required for a spring loaded cannon to launch you into space. A large force will accelerate you to escape velocity in less time than a small force. It may seem odd, but a small force applied long enough will allow you to escape the earth too, it will just take longer.